Monday, December 26, 2011

Transfinite Sequences

Sequence - Given a set X, a sequence is an ordered list of elements of X {x1, x2, x3, ...}. If the number of elements of the sequence is finite, the sequence is referred to as a "finite" sequence. A sequence can also be infinite.

More formally, a sequence is, strictly speaking, a function from some subset of the natural numbers N into some set X:

f:M \rightarrow X,\ M \subseteq \mathbb{N}

But wait a moment! What if we want for our function to range over a number of values which is greater than that in N? What if we want an uncountable sequence of numbers? What would that even mean?!

It turns out that mathematics does in fact give the machinery needed to handle this, although one must dig into set theory to find it. The notion of Transfinite Sequences allows for a sequence to be constructed whose length is that of any ordinal on any well-ordered set. If we accept the Axiom of Choice, then the Well Ordering Theorem holds (or visa versa; the axiom and the theorem are in fact logically equivalent) and we can produce a so-called "transfinite sequence" on any set, including the reals. The notions of Transfinite Induction and Transfinite Recursion also help to properly extend their respective notions which are classically limited to sets of countable order. Pretty cool stuff if you think about exactly what a sequence really is. Also sheds some light on just how extreme the consequences of the Axiom of Choice really are. On a side-note, I have only just begun reading on the topic of transfinite sequences today, so if something in this post is inaccurate, please feel free to correct me.